Geometric Quantization by Paths -- Part I: The Simply Connected Case
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911106164850688 |
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| author | Iglesias-Zemmour, Patrick |
| author_facet | Iglesias-Zemmour, Patrick |
| contents | For any connected and simply connected parasymplectic space $(\mathrm{X},ω)$ with group of periods $\mathrm{P}_ω\subsetneq \mathbf{R}$, we construct a prequantum groupoid $\pmb{\mathrm{T}}_ω$ as a diffeological quotient of the space $\mathrm{Paths}(\mathrm{X})$ of paths in $\mathrm{X}$. This object, built from the geometry of the classical system, serves as a unified structure for prequantization. The groupoid $\pmb{\mathrm{T}}_ω$ has $\mathrm{X}$ as its objects, and its space of morphisms $\mathcal{Y}$ carries a canonical left-right invariant $1$-form $\pmbλ$ whose curvature encodes $ω$. A key property is that the isotropy group $\pmb{\mathrm{T}}_{ω,x}$ at any point $x$, naturally arising as a quotient of the space of loops, is isomorphic to the torus of periods $\mathrm{T}_ω= \mathbf{R}/\mathrm{P}_ω$. Furthermore, the entire symmetry group $\mathrm{Diff}(\mathrm{X}, ω)$ acts as faithful automorphisms of $(\pmb{\mathrm{T}}_ω, \pmbλ)$ without central extensions at this level. Built within the framework of diffeology, this construction generalizes classical prequantization by applying to broad classes of spaces, including those with singularities or infinite-dimensional aspects, and by accommodating generalized (e.g., irrational) tori of periods. This paper focuses on the simply connected case; the construction will be extended to general diffeological spaces in a subsequent publication. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_11337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Quantization by Paths -- Part I: The Simply Connected Case Iglesias-Zemmour, Patrick Mathematical Physics Primary 53D50, 58A05, Secondary 22A22, 55R65, 58A10 For any connected and simply connected parasymplectic space $(\mathrm{X},ω)$ with group of periods $\mathrm{P}_ω\subsetneq \mathbf{R}$, we construct a prequantum groupoid $\pmb{\mathrm{T}}_ω$ as a diffeological quotient of the space $\mathrm{Paths}(\mathrm{X})$ of paths in $\mathrm{X}$. This object, built from the geometry of the classical system, serves as a unified structure for prequantization. The groupoid $\pmb{\mathrm{T}}_ω$ has $\mathrm{X}$ as its objects, and its space of morphisms $\mathcal{Y}$ carries a canonical left-right invariant $1$-form $\pmbλ$ whose curvature encodes $ω$. A key property is that the isotropy group $\pmb{\mathrm{T}}_{ω,x}$ at any point $x$, naturally arising as a quotient of the space of loops, is isomorphic to the torus of periods $\mathrm{T}_ω= \mathbf{R}/\mathrm{P}_ω$. Furthermore, the entire symmetry group $\mathrm{Diff}(\mathrm{X}, ω)$ acts as faithful automorphisms of $(\pmb{\mathrm{T}}_ω, \pmbλ)$ without central extensions at this level. Built within the framework of diffeology, this construction generalizes classical prequantization by applying to broad classes of spaces, including those with singularities or infinite-dimensional aspects, and by accommodating generalized (e.g., irrational) tori of periods. This paper focuses on the simply connected case; the construction will be extended to general diffeological spaces in a subsequent publication. |
| title | Geometric Quantization by Paths -- Part I: The Simply Connected Case |
| topic | Mathematical Physics Primary 53D50, 58A05, Secondary 22A22, 55R65, 58A10 |
| url | https://arxiv.org/abs/2508.11337 |